To get to the Train Exhibit in Luigi’s Mansion 2 and complete the mission “E-3: A Train to Catch,” follow these detailed steps:

### Step 1: Reassemble the Fan in the Inner Courtyard

1. Start at the Inner Courtyard: Begin your journey by reassembling the fan on the staircase. Press X to reassemble it.
2. Use Poltergust: Use your Poltergust to spin the fan so that the top of the staircase leads to the right wing.
3. Climb Up: Climb up the stairs and enter the East Corridor.

### Step 2: Navigate Through the East Corridor

1. Follow Coins: Follow the trail of coins to the Train Exhibit.
2. Greenie Encounter: A Greenie will slam the door on Luigi, and several Creepers will appear. Be cautious and watch your step.
3. Enter Study: After the ghosts are gone, head into the door beside the Train Exhibit to enter the Study.

### Step 3: Solve the Study Puzzle

1. Missing Book: There is a book missing from the shelf in the Study. Use your Dark-Light to reveal where it is.
2. Place Book: Place the book on the bookshelf to summon an Empowered Poltergeist.
3. Empowered Poltergeist: The Empowered Poltergeist will steal the book back before fleeing. The Professor will inform you that the ghost has moved to another part of the mansion.
4. Move Stairs: Go back to the Inner Courtyard and move the stairs to the left.
5. West Corridor: Make your way through the West Corridor and into the Aviation Exhibit.

### Step 4: Complete Tasks in the Aviation Exhibit

1. Diamond: Pull one of the sandbags on the hot air balloon in the Aviation Exhibit. An Empowered Hider and two Greenies will spring forth. Capture them and pull the other sandbag to reveal a gem.
2. Money Spots: Be on the lookout for gold spiders and mice in the West Corridor. In the Aviation Room, use your Poltergust on the propeller of the model plane to guide it around the room and get rewarded with cash.
3. Archives: Approach several closed boxes in the Archives. Open the bottom left, top right, top left, and bottom right to reveal a Goldie.

### Step 5: Prepare for the Train Exhibit

1. Balloon Berry: Grab one of the Balloon Berries from the Study.
2. Portal: Enter the portal to return to the Front Entrance. Float upwards and to the right to find a small landing with a gargoyle holding a blue book in his mouth.
3. Retrieve Book: Make the gargoyle drop the book and bring it back to its rightful place in the Dark Age Exhibit to find a gem.
4. Hidden Boo: Grab a Balloon Berry from the Study and go to the Dark Age Exhibit. Float to the upper level to find a gold Balloon Berry and start the next minigame. Grab all the coins before time runs out to reveal a chest with the Boo inside.
5. Money Spot: Take a Balloon Berry from the Study and go to the Ancient Exhibit to find several chests and a gold Balloon Berry. Grab the berry to start the minigame. Float against the current of coins to collect them in time and retrieve your prize.

### Step 6: Enter the Train Exhibit

1. Book Placement: Take the book that you won back from the Poltergeist into the Study and place it in the bookshelf to open a path to the Train Exhibit.
2. Dark-Light: Use your Dark-Light on the floor to reveal a missing set of tracks, which brings a large train into the room.
3. Train Ride: Step onto the train for the ride of your life and collect the gem for your troubles.
4. Money Spot: Flash the giant model train on the left side of the room for some quick cash!

### Step 7: Prepare for the Boss Battle

1. Ambush: Prepare to be ambushed by Empowered Hiders and Empowered Sneakers. Keep your eyes out for projectiles and Sneakers at all times.
2. Strategy: Try to get the Hiders on one side of the room while you lure the Sneaker to the other. When they are finished, the room will be yours and the mission will be complete.

### Step 8: Complete the Boss Battle

1. Big Boo Encounter: The exhibit seems a bit too quiet… shine your Dark-Light around the train to reveal a giant drill attached to the front of the vehicle. Discovering the drill will encourage Big Boo to show himself.
2. Boss Battle Strategy:
– Dodge Big Boo: Big Boo cannot be attacked directly. Shining your flashlight in his eyes will make him hide his face temporarily.
– Use Train Drill: Wait until Big Boo knocks himself out and then suck his tongue into your Poltergust. Aim him in the direction of the drill on the train. If he hits the drill, he explodes into lots of little Boos.
– Capture Little Boos: The little Boos are just like the others you have faced and can be stunned by the Dark-Light. Aim these smaller Boos at the open train compartments. Once you shoot a Boo into one of the compartments, the lid will close and seal it in. You can only lock one Boo in each compartment.
– Repeat Process: As you progress through the battle and have fewer compartments to fill, it starts getting harder and harder to aim the Boos. Patience is key in this fight. Trying to shoot Boos at random will seldom pay off. Instead, grab hold of a Boo, aim it toward the tracks, and wait for the train to pass at just the right spot before launching it.
– Convergence: The little Boos don’t hang around forever. If you take too long to take your shot, the Boos will converge and reassemble into Big Boo again. When that happens, repeat the process of flinging Big Boo into the drill and aiming the little Boos at the train compartments until there are none left.

By following these steps, you will successfully navigate through the Train Exhibit and complete the mission “E-3: A Train to Catch” in Luigi’s Mansion 2.